Limit theorems for BSDE with local time applications to nonlinear - - PowerPoint PPT Presentation

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDE with local time applications to non–linear PDE

M’hamed Eddahbi

Universit´ e Cadi Ayyad, Facult´ e des Sciences et Techniques D´ epartement de Math´ ematiques, B.P. 549, Marrakech, Maroc.

´ Equipe d’Analyse Math´ ematique et Finance Joint work with Y. Ouknine Based on Stoc. Stoc. Reports 73, (1-2), 159–179, 2002 New advances in Backward SDEs for financial engineering applications Tamerza Palace, Tunis, October, 25–28, 2010

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Introduction

We prove limit theorems for solutions of BSDEs with local time.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Introduction

We prove limit theorems for solutions of BSDEs with local time. Those limit theorems will permit us to deduce that any solution of that equation is the limit in a strong sense of a sequence of semi–martingales which are solutions of ordinary BSDE.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Introduction

We prove limit theorems for solutions of BSDEs with local time. Those limit theorems will permit us to deduce that any solution of that equation is the limit in a strong sense of a sequence of semi–martingales which are solutions of ordinary BSDE. comparison theorem for BSDE involving measures is discussed. As an application we obtain, with the help of the connection between BSDE and PDE, some corresponding limit theorems for a class of singular non–linear PDE and a new probabilistic proof of the comparison theorem for PDE.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Introduction

BSDEs : Bismut 1976 in the linear case.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Introduction

BSDEs : Bismut 1976 in the linear case. Non–linear BSDE : Pardoux and Peng in 1990.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Introduction

BSDEs : Bismut 1976 in the linear case. Non–linear BSDE : Pardoux and Peng in 1990. Motivations : BSDE and mathematical finance (El Karoui et al. 1997), Probabilistic interpretation of PDE Pardoux-Peng, Stochastic differential games and stochastic control : Hamad` ene-Lepeltier 1995 etc Quadratic BSDE (Imkeller, CIRM 2006)

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

Consider the following particular BSDE Yt = ξ + T

t

f (Ys)Z 2

s ds −

T

t

ZsdWs. (2.1) From the equality d Y , Y t = Z 2

t dt and from occupation time

formula, we have, for any bounded measurable function f t f (Ys)Z 2

s ds =

∞

−∞

La

t(Y )f (a)da.

Set ν(da) = f (a)da, then (2.1) takes the form Yt = ξ +

• R

(La

T(Y ) − La t(Y )) ν(da) −

T

t

ZsdWs (2.2)

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

The process La

t(Y ) is the local time of the continuous

semi-martingales Y and can be expressed by Tanaka’s formula as La

t(Y ) = |Yt − a| − |Y0 − a| −

t sgn(Ys − a)dYs and sgn(x) =    1 for x > 0 for x = 0 −1 for x < 0. It is proved by Dermoune et al. ’99 that there exists an adapted couple (Y , Z) solution to equation (2.2) under the following conditions : (H1) The r.v. ξ belongs to L2(Ω, FT, P). (H2) The measure ν is bounded and |ν({x})| < 1, ∀ x in R.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

Our aim in this talk is to prove some limit theorems for the class of BSDE of the form (2.2), that are some kind of the stability properties for BSDEs. We show that a solution to (2.2) can be obtained as a limit of sequence of solution to (2.1). To prove a comparison theorem for the above singular BSDE, As application : limit theorems in the monotone case. We deduce limit theorems for a class of non–linear PDEs involving the square of the gradient and a comparison theorem is discussed for this PDEs.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

The main tool to study the BSDE (2.2) is the Zvonkin’s transformation . Let us set fν(x) = exp (2νc((−∞, x]))

• y≤x

1 + ν(({y})) 1 − ν(({y}))

• where νc is the continuous part of the measure ν.

If f is of bounded variation (increasing in our case), f (x−) will denote the left limit of f at a point x and f ′(dx) will be the bounded measure associated with f .

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

It is well known that the function (since ν is bounded) that fν(·) is increasing, right continuous and satisfies 0 < m ≤ fν(x) ≤ M ∀ x ∈ R for some constants m, M. Moreover fν satisfies f ′

ν(dx) − {fν(x) + fν(x−)} ν(dx) = 0.

Set Fν(x) = x fν(y)dy and gν(x) = fν(F −1

ν (x)).

The functions Fν and F −1

ν

are Lipschitz functions.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

Let M2

T(R × Rd) denote the space of Ft–prog. meas. proc.

(Y , Z) satisfying (??) Proposition (Y , Z) ∈ M2

T(R × Rd) solves (2.2) iff

• ˜

Y , ˜ Z

• =
• Fν(Y ), Z

2 {fν(Y ) + fν(Y −)}

• solves ˜

ξ = Fν(ξ) the BSDE ˜ Yt = ˜ ξ − T

t

˜ ZsdWs, (2.3)

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

• Proof. The proof is based on Tanaka’s formula to Fν(Yt) with the

symmetric derivative of the convex function Fν instead of its left derivative. Remark Stroock and Yor (1981), Le Gall ’84) and Rutkowski ’90 have already used the transformation Fν to study the SDE Xt = x + t σ(Xs)dWs +

• R

La

t(X)ν(da).

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

Theorem Under the assumptions (H1) and (H2), there exists a unique solution (Y ν, Z ν) belonging to M2

T(R × Rd) for the equation

(2.2). Moreover Y ν

t = F −1 ν

(E [Fν(ξ) / Ft ]) , 0 ≤ t ≤ T.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

Example Let ν = αδ, where |α| < 1. Then fν(x) = 1 for x < 0 and fν(x) = 1+α

1−α for x ≥ 0. The function Fν(x) = x for x < 0 and

Fν(x) = 1+α

1−αx for x ≥ 0. The solution of the BSDE

Yt = ξ + αL0

T(Y ) − αL0 t (Y ) −

T

t

ZsdWs, where ξ ∈] − ∞, 0[ or ξ ∈ [0, ∞[ is given by Yt = E [ξ /Ft ] , and L0

t (Y ) = 0 for all 0 ≤ t ≤ T.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDE with local time

Remark In the case where ν is a non–necessary bounded measure on R which is diffuse and σ–finite, the associated function fν(x) = exp(2ν((−∞, x])) is positive, continuous and non necessary bounded function. Hence the function Fν(x) is only locally Lipschitz, however if ξ and Fν(ξ) are square integrable random variables then the BSDE (2.2) has a unique solution which is given by Y ν

t = F −1 ν

(E [Fν(ξ) /Ft ]) .

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

Let νn(da), n = 1, 2, . . . be a sequence of Radon measures and ξn a sequence of random variables in L2(Ω, FT, P). Suppose that there exist two positive constants ε, M such that : |νn| (R) ≤ M ∀ n ≥ 1, |νn({x})| ≤ ε < 1 ∀ n ≥ 1, ∀ x ∈ R. Let (Y n, Z n) be the solution of Yt = ξn +

• R

(La

T(Y ) − La t(Y )) νn(da) −

T

t

ZsdWs.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

Assume that ξn − →n→∞ ξ in L2(Ω, FT, P). Assume further that there exist a function f BV such that : lim

n→+∞

L

−L

|fνn − f |2 (x) dx = 0 for all L > 0, ν(da) = f ′ (da) f (a) + f (a−)· Then lim

n→+∞ E sup 0≤t≤T

|Y n

t − Y ν t |2 + E

T |Z n

s − Z ν s |2 ds = 0

(3.4) where (Y ν, Z ν) is the unique solution to the BSDE equation : Yt = ξ +

• R

(La

T(Y ) − La t(Y )) ν(da) −

T

t

ZsdWs.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

• Proof. We shall use the following notations :

fνn(x) = exp(2νc

n((−∞, x]))

• y≤x

1 + νn(({y})) 1 − νn(({y}))

• Fνn(y) =

y fνn(x)dx and F(y) = y f (x)dx. By Theorem 1, it holds Y n

t

= F −1

νn (E [Fνn(ξn) /Ft ])

0 ≤ t ≤ T. Y ν

t

= F −1 (E [F(ξ) /Ft ]) 0 ≤ t ≤ T.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

The convergence of fνn to f in L2

loc(R) implies that Fνn converges

to F uniformly on compact sets and then, using a truncating argument, Fνn(ξn) converges to F(ξ). It follows that Yt

n := E[Fνn(ξn)/Ft] converges to E[F(ξ)/Ft] =: Yt ν in L2(Ω). It

is, trivial to see that F −1

νn converges to F −1 uniformly on compact

sets and so Y n

t = F −1 νn (Yt n) converges to F −1(Yt ν) = Y ν t . Hence

Esup0≤t≤T |Y n

t − Y ν t | tends to zero when n goes to infinity, and

using the isometry property, one can see that E T

0 |Z n s − Z ν s |2 ds

converges to zero when n tends to infinity.

• M’hamed Eddahbi

Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

Remark Let ξn = ξ for all n, νn(dx) = fn (x) dx where fn (x) ≥ 0 ;

• fn (x) dx = 1 and supp (fn) = [− 1

n, 1 n].

Let us consider the BSDE Y n

t = ξ +

T

t

fn(Y n

s ) (Z n s )2 ds −

T

t

Z n

s dWs,

then the last theorem implies the convergence of Y n

· to the unique

solution of the BSDE Yt = ξ + 1 − e2 1 + e2

• L0

T(Y ) − L0 t (Y )

• −

T

t

ZsdWs.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

If νn converges to a measure ν, then in general Y νn does not converges to Y ν. We replace the convergence of measures νn by the convergence of its associated function fνn. In the sequel M (R) will denote the space of all bounded measure

• n R such that :

|ν({x})| < 1 ∀ x ∈ R. Let ν be in M (R) . We define ν = |νc(R)| + 1 2

• y
• 1 + ν({y})

1 − ν({y})

• ·

Note that ν = var 1 2 log (fν)

• where, var, denotes the total variation.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

In the space Let M2

T we define the distance d[., .] given by :

d

• (Y , Z) ,
• Y ′, Z ′

=

• E sup

0≤t≤T

• Yt − Y ′

t

• 2 + E

T

• Zs − Z ′

s

• 2 ds

1 2 · Theorem Let C be a fixed constant. Then, K = {(Y ν, Z ν) : ν ≤ C} is a compact set for the topology induced by d[·, ·]. The set of all (Y ν, Z ν) belonging to K such that ν is absolutely continuous with respect to Lebesgue measure is dense in K.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

• Proof. Let νn be a sequence in M (R) such that νn ≤ C.

Since the total variation of the fνn’s are uniformly bounded, we can find a function f of bounded variation and a subsequence (fνnk ) such that : fνnk (x) − →f (x) as k − → +∞, for all x ∈ R\Df where, Df , is at most countable. Set ν(da) = f ′ (da) f (a) + f (a−)· Then the first limit Theorem implies that : d [(Y νnk , Z νnk ) , (Z ν, Z ν)] − →0 when k − → +∞.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

It remains to prove that ν ≤ C. Note that f satisfies the same equation as fν, then, there exist λ > 0 such that f = λfν. Hence ν = var 1 2 log (fν)

• =

var 1 2 log (f )

• ≤

lim sup

n→+∞

var 1 2 log (fνn)

• ≤ C.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Limit theorems for BSDEs

Let us prove the second point ; Let ν ∈ M (R) and θn an approximation of the identity. Set fn = fν ∗ θn and gn = f ′

n

2fn · Let (Y n, Z n) be the unique solution of the following BSDE Y n

t = ξ +

T

t

gn(Y n

s ) (Z n s )2 ds −

T

t

Z n

s dWs·

Using Theorem 3.4, it is easy to see that : d [(Y n, Z n) , (Y ν, Z ν)] − → 0 as n − → +∞.

• M’hamed Eddahbi

Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

Lepeltier and San Martin consider BSDEs with terminal data ξ ∈ L∞(Ω, FT, P), and gave a comparison theorem for BSDE with parameter (f , ξ) i.e. Yt = ξ + T

t

f (Ys)Z 2

s ds −

T

t

ZsdWs. In the following theorem, we prove a general comparison theorem, without boundedness of the terminal value of the BSDE. As a byproduct, we obtain the comparison theorem, for the standard BSDE under fairly weak conditions on the coefficients.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

Theorem Let ν, µ be in M (R). Let (Y ν, Z ν), (Y µ, Z µ) be two processes such that : Y ν

t

= ξ +

• R

(La

T(Y ν) − La t(Y ν)) ν(da) −

T

t

Z ν

s dWs,

Y µ

t

= ξ′ +

• R

(La

T(Y µ) − La t(Y µ)) µ(da) −

T

t

Z µ

s dWs.

Assume that ξ ≥ ξ′ a.s. and the measure ν − µ is positive. Then Y ν

t ≥ Y µ t for all t P–a.s.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

• Proof. Let us first recall Tanaka’s formula. Since Fµ is a convex

function, then Fµ(Y ν

T)

= Fµ(Y ν

t ) +

T

t

1 2 (fµ(Y ν

s ) + fµ(Y ν s −)) dY ν s

+1 2

• R

(La

T(Y ν) − La t(Y ν)) f

′

µ (da)

hence Fµ(ξ) = Fµ(Y ν

t ) + (MT − Mt)

−1 2

• R

{fµ(a) + fµ(a−)} (La

T(Y ν) − La t(Y ν)) (ν − µ) (da)

where M· is a square integrable martingale.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

Since the function a → [fµ(a) + fµ(a−)](La

T(Y ν) − La t(Y ν)) is

positive, and Fµ is an increasing function, then Fµ (Y ν

t ) ≥ E

• Fµ(ξ′) /Ft
• and

Y ν

t ≥ F −1 µ

• E
• Fµ
• ξ′

/Ft

• = Y µ

t .

• M’hamed Eddahbi

Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

An immediate consequence of the above comparison result is the Corollary Let (νn)n≥1 be an sequence of measures such that supn≥1 νn < +∞ and fνn increases to a BV function f . If ξn increases to ξ ∈ L2(Ω, FT, P) as n → ∞. Then d[(Y ν, Z ν) − (Y n, Z n)] → 0 where ν(da) = f ′ (da) f (a) + f (a−)· and (Y n, Z n), (Y ν, Z ν) solves the corresponding BSDEs

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

Corollary Let (f 1, ξ1) and (f 2, ξ2) be two parameters of BSDE, and let

• Y 1, Z 1

and

• Y 2, Z 2

be associated solution. Suppose that : ξ1 ≤ ξ2 a.s. and f 1(y) ≤ f 2(y) for almost all y. Then for all t ∈ [0, T], we have Y 1

t ≤ Y 2 t a.s.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

Corollary Let (f 1, ξ1) and (f 2, ξ2) be two parameters of BSDE, and let

• Y 1, Z 1

and

• Y 2, Z 2

be associated solution. Suppose that : ξ1 ≤ ξ2 a.s. and f 1(y) ≤ f 2(y) for almost all y. Then for all t ∈ [0, T], we have Y 1

t ≤ Y 2 t a.s.

As a consequence of the above results, we have obtained an interesting limit theorem for generalized BSDE in monotonic case.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

Theorem Let (νn)n≥1 be an increasing sequence of measures such that supn≥1 νn < +∞, assume ξn increases to ξ ∈ L2(Ω, FT, P) as n tends towards infinity. Then lim

n→+∞ E sup 0≤t≤T

|Y ν

t − Y n t |2 +

T |Z ν

s − Z n s |2 ds = 0,

where (Y ν, Z ν) is the unique solution of the BSDE Y ν

t = ξ +

• R

(La

T(Y ν) − La t(Y ν)) ν(da) −

T

t

Z ν

s dWs,

and ν = supn≥1 νn.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

• Proof. For any measurable set, we have ν (A) = supn≥1 νn (A), it

follows from the bound supn≥1 νn < +∞, that ν is a bounded measure. Set Fn(y) := Fνn(y) and F(y) := Fν(y). Then Fn(·) is increasing and converges to the continuous function F(·), hence by Dini’s theorem this convergence is uniform. By the comparison Theorem 3, the sequence Y n

t is increasing. Set

Y ν

t =

lim

n→+∞Y n t ,

hence Fn(Y n

t ) converges to F(Y ν t ).

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

But Fn (Y n

t ) = E [Fn(ξn) /Ft ]

0 ≤ t ≤ T. and |Fn (ξn)| ≤

• ξ1

+ |ξ|

• exp (2|ν|(R)) .

Then passing to the limit, using dominated convergence theorem for conditional expectation, it holds that Y ν

t = F −1 (E [F(ξ) /Ft ])

0 ≤ t ≤ T.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorems for BSDEs

By Theorem 1, (Y ν, Z ν) is the unique solution of the BSDE Y ν

t = ξ +

• R

(La

T(Y ν) − La t(Y ν)) ν(da) −

T

t

Z ν

s dWs.

We deduce from Burkholder–Davis–Gundy inequality, that lim

n→+∞ E sup 0≤t≤T

|Y ν

t − Y n t |2 = 0,

using the transformation Fν and the isometry property, we get lim

n→+∞ E

T |Z ν

s − Z n s |2 ds = 0.

• M’hamed Eddahbi

Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

This section is devoted to limit theorems for PDE that can be deduced from the above limit theorems for BSDE using the connection between these different kind of equations. Let {X x,t

s

: 0 ≤ t ≤ s ≤ T} be the unique solution of the stochastic differential equation X x,t

s

= x + s

t

b(X x,t

r

)dr + s

t

σ(X x,t

r

)dWr, where the coefficients b and σ are globally Lipschitz.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

Let ν be a measure on R and satisfy the assumption (H2), we consider the singular non–linear Cauchy problem ∂u ∂t = Lu − 1 2σ2(x) ∂Fν(u) ∂x 2 Fν(u)∗ d2F −1

ν

d2x 2 u(0, x) = g(x), x ∈ R,    (5.5) where g is a continuous real valued function with polynomial growth and L is the infinitesimal generator of the diffusion process {X x,t

s

: 0 ≤ t ≤ s ≤ T} and π∗(φ) stands for the pullback of the distribution φ by π.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

In the case where the convex function Fν is twice continuously differentiable, the equation (5.5) takes the form ∂u ∂t = Lu + σ2(x) F ′′

ν (u)

2F ′

ν(u)

∂u ∂x 2 u(0, x) = g(x), x ∈ R.    (5.6) This situation corresponds to the case where ν << dx. Let {Y x,t

s

: s ∈ [t, T]} be the unique solution to BSDE Y x,t

s

= g(X x,t

T ) +

T

s

F ′′

ν

2F ′

ν

(Y x,t

r

)

• Z x,t

r

2 ds − T

s

Z x,t

r

dWr, t ≤ s Then Y x,t

t

is a viscosity solution to the equation (5.6).

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

Now, let {(Y x,t

s

, Z x,t

s

) : s ∈ [t, T]} be the unique solution to the singular BSDE Y x,t

s

= g(X x,t

T )+

• R
• La

T

• Y x,t

− La

s

• Y x,t

ν(da)− T

s

Z x,t

r

dWr. With the help of the transformation Fν one can see that Y x,t

t

is a viscosity solution to the equation (5.5). Let us now go back to the corresponding limit theorems.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

Theorem Let νn(da), n = 1, 2, . . . be a sequence of Radon

• measures. Suppose that there exist two positive constants ε, M

such that : |νn| (R) ≤ M ∀ n ≥ 1, |νn({x})| ≤ ε < 1 ∀ n ≥ 1, ∀ x ∈ R. and there exist a function f BV such that : L

−L

|fνn − f |2 (x) dx− →0 as n → +∞ for all L > 0,

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

Set ν(da) = f ′ (da) f (a) + f (a−) and F(x) := x f (y)dy· If un(t, x) and u(t, x) denote respectively the unique solution of the PDE (5.5) with νn respectively with ν. Then un(t, x) converges to u(t, x) as n tends to ∞ for any (t, x) ∈ [0, T] × R.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

• Proof. For any t ∈ R+, we let {Y x,t,n

s

: t ≤ s ≤ T} and {Y x,t

s

: t ≤ s ≤ T} be respectively the solution of the BSDE Y x,t,n

s

= g(X x,t

T )+

• R
• La

T

• Y x,t,n

− La

s

• Y x,t,n

νn(da)− T

s

Z x,t,n

r

dWr and Y x,t

s

= g(X x,t

T )+

• R
• La

T

• Y x,t

− La

s

• Y x,t

ν(da)− T

s

Z x,t

r

dWr. If we set ˜ u := Fν(u), then equation (5.5) becomes ∂˜ u ∂t (t, x) = L˜ u(t, x) ˜ u(0, x) = Fν(g(x)), x ∈ R.

• M’hamed Eddahbi

Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

Therefore the process {˜ u(s, X x,t

s

) : t ≤ s ≤ T} is the unique solution to the BSDE ˜ Y x,t

s

= Fν

• g(X x,t

T )

• −

T

s

˜ Z x,t

r

dWr, hence Y x,t

s

= F −1

ν ( ˜

Y x,t

s

) = F −1

ν (˜

u(s, X x,t

s

)) = u(s, X x,t

s

), in particular un(t, x) = Y x,t,n

t

and Y x,t

t

:= uν(t, x), then by virtue of the previous results, un(t, x) and uν(t, x) are respectively the unique viscosity solution to (5.5) with νn respectively ν. So Theorem 3.4 implies that lim

n→+∞ E sup t≤s≤T

• Y x,t,n

s

− Y x,t

s

• = 0

which implies that un(t, x) converges to u(t, x) as n tends to ∞. The convergence is uniform on compacts by continuity.

• M’hamed Eddahbi

Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

Let uhn be the unique solution of the following PDE ∂u ∂t (t, x) = Lu(t, x) + σ2(x)hn(u(t, x)) ∂u ∂x (t, x) 2 u(0, x) = g(x), x ∈ R.    (5.7) The following theorem gives the relative compactness of the family {uν : ν ≤ C} and states that a solution to equation (5.5) is a limit of sequence of solution to the equation (5.7).

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

Theorem Let C be a fixed constant. Then, K = {uν : ν ≤ C} is a compact set for the topology induced by uniform convergence. The set of all uν belonging to K such that ν is absolutely continuous with respect to Lebesgue measure is dense in K.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

Theorem Let C be a fixed constant. Then, K = {uν : ν ≤ C} is a compact set for the topology induced by uniform convergence. The set of all uν belonging to K such that ν is absolutely continuous with respect to Lebesgue measure is dense in K.

• Proof. The proof of the first part is an immediate consequence of

the connection between BSDEs and PDEs and Theorem 2. Let us prove the second part ; Let ν be in M (R) and θn be an approximation of the identity, we set fn = fν ∗ θn and gn = f ′

n

2fn .

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Applications to non–linear PDE

Let ugn be the unique solution of the PDE (5.7). Using BSDE representation we get : lim

n→+∞ ugn − uν∞ = 0.

• We use the connection between BSDEs and PDEs to give a

probabilistic proof to a comparison theorem for non–linear PDE. Theorem Let g1 and g2 be two functions such that g1(x) ≤ g2(x), ∀ x ∈ R. Let ν1 and ν2 be in M (R) such that the measure ν2 − ν1 ≥ 0. If u1 and u2 are the solutions to the PDE (5.5) corresponding to ν1 and ν2. Then u1(t, x) ≤ u2(t, x).

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorem for PDEs.

• Proof. We can write u1(t, x) = Y x,t,1

t

and u2(t, x) = Y x,t,2

t

, where {(Y x,t,i

s

, Z x,t,i

r

) : t ≤ s ≤ T} is the unique solution to the BSDE Ys = gi(X x,t

T )+

• R

(La

T (Y ) − La s (Y )) νi(da)−

T

s

ZrdWr, i = 1, 2. Now, from Theorem 3, we have Y x,t,1

s

≤ Y x,t,2

s

for all t ≤ s ≤ T, in particular Y x,t,1

t

≤ Y x,t,2

t

.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorem for PDEs.

Remark Using the same argument as in Corollary 1 and the comparison Theorem 6 one can obtain the corresponding limit theorem in the monotone case for PDE. Theorem 6 implies the uniqueness property for a class of non–linear PDEs (at least of the form (5.5)).

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorem for PDEs.

Example Consider the non-linear Cauchy problem ∂u ∂t (t, x) = 1 2∆u(t, x) − 1 2 ∂u ∂x (t, x) 2 + k(x), (5.8) where u(0, x) = g(x), x ∈ R and k : R → [0, +∞) is continuous. If g = 0, the unique solution, is given for all (t, x) ∈ [0, +∞[×R by : u(t, x) = − log E

• exp
• −

t k(x + Ws)ds

• .

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorem for PDEs.

In the case where k = 0 the equation (5.8) is a particular equation

• f (5.7) which corresponds to the case where σ = 1, b = 0 and

h = − 1

2, hence we can use BSDE to construct a solution to the

equation (5.8), more precisely, let {Y x,t

s

: t ≤ s ≤ T} be the unique solution to the BSDE Y x,t

s

= g(X x,t

T ) −

T

s

1 2

• Z x,t

r

2 dr − T

s

Z x,t

r

dWr (5.9) where X x,t

T

= x + 1

2(BT − Bt).

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Comparison theorem for PDEs.

It is clear that the hypotheses of the Remark 2 are satisfied since the Brownian motion has exponential finite moment, and hence the equation (5.9) has a unique solution and u(t, x) = Y x,t

t

is the viscosity solution to the equation (5.8) with u(0, ·) = g(·). In the case where k = 0, the BSDE associated to the equation (5.8) is the following Y x,t

s

= g(X x,t

T )+

T

s

k(X x,t

r

)dr − T

s

1 2

• Z x,t

r

2 dr − T

s

Z x,t

r

dWr, consequently the function u(t, x) = Y x,t

t

is the unique viscosity solution to the equation (5.8).

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDEs with reflexion

Let us now assume that ν be nonnegative σ–finite measure on R. We want to solve Yt = ξ +

• R

(La

T(Y ) − La t(Y )) ν(da) −

T

t

Zs · dWs. (5.10) Set α = ν({x0}), such that |α| ≥ 1. The equation (5.10) is equivalent to Yt = ξ +

• R

(La

T(Y ) − La t(Y )) µ(da)

(5.11) +α

• Lx0

T (Y ) − Lx0 t (Y )

• −

T

t

Zs · dWs (5.12) where µ(da) = ν(da) − αδx0(da), |µ({x})| < 1 for all x ∈ R.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDEs with reflexion

Lemma The equation (5.11) is equivalent to

• Yt =

ξ − T

t

• Zs · dWs + α
• LFµ(x0)

T

( Y ) − LFµ(x0)

t

( Y )

• .

(5.13)

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDEs with reflexion

• Proof. Let us now use the transformation Fµ, using Tanaka

formula we obtain Fµ(YT) = Fµ(Yt) + T

t

1 2 (fµ(Ys) + fµ(Ys−)) dYs +1 2

• R

(La

T(Y ) − La t(Y )) f

′

µ (da)

= Fµ(Yt) + T

t

1 2 (fµ(Ys) + fµ(Ys−)) Zs · dWs +LFµ(x0)

T

(Fµ(Y )) − LFµ(x0)

t

(Fµ(Y )).

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDEs with reflexion

Set αi = ν({xi}), such that |αi| ≥ 1 for i ∈ I, where I is at most a countably subset of indices. The equation (5.10) is equivalent to Yt = ξ +

• R

(La

T(Y ) − La t(Y )) µ(da)

+

• i∈I

αi

• Lxi

T(Y ) − Lxi t (Y )

• −

T

t

Zs · dWs where µ(da) = ν(da) −

i∈I αiδxi(da). Now we may assume

|µ({x})| < 1 for all x ∈ R.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDEs with reflexion

Lemma The equation (5.11) is equivalent to Yt = Fµ(ξ) − T

t

Zs · dWs +

• i∈I

αi

• LFµ(xi)

T

(Y ) − LFµ(xi)

t

(Y )

• .

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDEs with reflexion

To study the BSDE of the form equation (5.11) it suffices to solve the following BSDE Yt = ξ − T

t

Zs · dWs + α

• Lx0

T (Y ) − Lx0 t (Y )

• .

(5.14) Without loss of generality we may assume that α ≥ 1. The problem (5.14) is a classical reflected BSDE at the deterministic point x0. Proposition If ξ ≤ x0 a.s. or ξ ≥ x0 a.s. such that ν({x0}) = α. Then the equation (5.14) has a unique solution.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDEs with reflexion

Proposition Let x1 and x2 be two real numbers such that ν({x1}) = α1, ν({x2}) = α2 and x1 < x2. If x1 ≤ ξ ≤ x2 a.s. Then the equation Yt = ξ− T

t

Zs·dWs+α1

• Lx1

T (Y ) − Lx1 t (Y )

• +α2
• Lx2

T (Y ) − Lx2 t (Y )

• has a unique solution. Moreover x1 ≤ Yt ≤ x2 a.s. and

Lx1

t (Y ) = Lx2 t (Y ) = 0, for all t ∈ [0, T].

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDEs with reflexion

Proof of Proposition 2. Consider the following reflected BSDE Yt = ξ − T

t

Zs · dWs + KT − Kt, where {Kt : t ∈ [0, T]} is an increasing such that K0 = 0. The reflected BSDE has a unique solution. (Yt = E(ξ/Ft), Zt, 0) is a

• solution. ξ ≤ x0 a.s. implies that Yt ≤ x0 a.s. Consequently

Lx0

t (Y ) = 0 for all t ∈ [0, T], indeed by Tanaka’s formula

(ξ − x0)+ = (Yt − x0)+ + T

t

1{Ys>x0}dYs + 1 2

• Lx0

T (Y ) − Lx0 t (Y )

• But

T

t 1{Ys>x0}dYs = 0, hence Lx0 T (Y ) = Lx0 t (Y ) = Lx0 0 (Y ) = 0.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

BSDEs with reflexion

The Proposition 3 corresponds to a two barrier reflected BSDE whose proof is similar to that of Proposition 2. Proposition OPEN PROBLEM : If P (ξ ≤ x0) P (ξ ≥ x0) > 0 such that ν({x0}) = α. Then equation (5.14) has no unique solution.

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea

Introduction BSDE with singular drift Limit theorems for BSDEs Comparison theorems for BSDEs Applications to non–linear PDE

Thank you for your attention

M’hamed Eddahbi Limit theorems for BSDE with local time applications to non–linea